Possebon needed oxygen and extensive treatement before being stretchered off in the second half of United's 3-1 League Cup win over Boro on Tuesday night following a studs-up challenge that earned Pogatetz a straight red card.
The Boro captain compounded his action by initially protesting his innocence, infuriating United boss Sir Alex Ferguson in the process.
On Wednesday however he was contrite. "Firstly, I have to say that I went for the tackle because I thought I could win the ball - I now accept, however that I misjudged the challenge," Pogatetz told his club's website.
"Immediately after the incident I did not think it was a red card offence because I did win the ball,"he added. "But now I that I have had the chance to see the tackle on television, I know the ref was right to send me off.
"I am going to speak to Rodrigo by telephone and I will apologise for hurting him. I hope that he will make a quick recovery and that he will not be out for long."
United confirmed earlier on Wednesday that Possebon had not broken his right leg, as first feared, but it remains to be seen how long the Brazilian will take to recover from a tackle that left him with a deep gash just below his right knee.
Possebon, 19, was making his first start for United in a competitive match.